Problem 4 Suppose R³ T →R3 is a linear transformation with ¹8-8-8-8-8-3 T = 0 T -1 and T 0 Find the matrix of T. (Hint: If S is the matrix you're supposed to find, write the initial data as SA= B for matrices A and B obtained by stacking together the input and out vectors respectively. Then you can recover S as S = BA ¹.)
Problem 4 Suppose R³ T →R3 is a linear transformation with ¹8-8-8-8-8-3 T = 0 T -1 and T 0 Find the matrix of T. (Hint: If S is the matrix you're supposed to find, write the initial data as SA= B for matrices A and B obtained by stacking together the input and out vectors respectively. Then you can recover S as S = BA ¹.)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![**Problem 4**
Suppose \( \mathbb{R}^3 \xrightarrow{T} \mathbb{R}^3 \) is a linear transformation with
\[
T \begin{bmatrix} 1 \\ -1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \quad T \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}, \quad \text{and} \quad T \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 1 \\ -1 \end{bmatrix}.
\]
Find the matrix of \( T \).
*(Hint: If \( S \) is the matrix you're supposed to find, write the initial data as \( SA = B \) for matrices \( A \) and \( B \) obtained by stacking together the input and output vectors respectively. Then you can recover \( S \) as \( S = BA^{-1} \).)*
**Explanation:**
Given three transformation equations from vectors in \( \mathbb{R}^3 \), the task is to determine the matrix \( S \) of the linear transformation \( T \). Use the provided hint to construct matrices \( A \) and \( B \), where \( A \) consists of input vectors and \( B \) consists of corresponding output vectors. Compute the inverse of matrix \( A \), and multiply it (from the right) by \( B \) to obtain matrix \( S \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5dabe4e8-4d41-42d6-aeb9-4a7fca8c4b16%2Fcafd9f90-dd0a-4bcb-b6c5-91033c22cca1%2Fof9fwx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4**
Suppose \( \mathbb{R}^3 \xrightarrow{T} \mathbb{R}^3 \) is a linear transformation with
\[
T \begin{bmatrix} 1 \\ -1 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}, \quad T \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 2 \\ 4 \\ 1 \end{bmatrix}, \quad \text{and} \quad T \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 6 \\ 1 \\ -1 \end{bmatrix}.
\]
Find the matrix of \( T \).
*(Hint: If \( S \) is the matrix you're supposed to find, write the initial data as \( SA = B \) for matrices \( A \) and \( B \) obtained by stacking together the input and output vectors respectively. Then you can recover \( S \) as \( S = BA^{-1} \).)*
**Explanation:**
Given three transformation equations from vectors in \( \mathbb{R}^3 \), the task is to determine the matrix \( S \) of the linear transformation \( T \). Use the provided hint to construct matrices \( A \) and \( B \), where \( A \) consists of input vectors and \( B \) consists of corresponding output vectors. Compute the inverse of matrix \( A \), and multiply it (from the right) by \( B \) to obtain matrix \( S \).
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